A small number of niche noble companies specialize in collecting trash off of city streets and repacking it for later use. There are deals between apartment buildings and these companies to force tenants to give up their trash in a timely fashion. Huge dumpsters exist around Stem City for public disposal. Bimonthly, specialized city Lawairs fly out from the Nexus to these sites to seal and then extract the storage units from the ground. They'll fly to Stem Island's edge and dump the contents into the abyss before placing the container back.
Dead bodies go through an interesting process. Wealthy nobles who can afford it have their families buried in the noble neighborhood's remote cemetery. If one is a nobody in Stem City, a morgue company will gladly take the bodies off your hands (or off the street if they find them, which they very often do). Once in the company's hands, these bodies will have anything valuable stripped from them, such as remaining clothes, mechanical implants, spare organs, and teeth.
After the methodical extraction of any goods, these companies give detailed reports of the person's figure to Stem Tower. A system of trained Lawmen will cross match the person's references from these reports to Stem City's Markcard database and cancel the continued use of their corresponding Markcard. If the person can't be identified, a common occurrence, then no extra action is taken.
Once a quota of corpses is reported, Lawairs will be sent to extract a large container full of bodies. They will fly to the city's edge and dump the remains into the abyss before returning the container to the owner.
In the Outlands, trash is just thrown to the ground whenever and wherever, and dead bodies are given a shallow burial in the Fungal Plains' soil.